What is this about? I just started playing and saw this in the menu. Also, any other ui or basic gameplay knowledge/tips you have would be great. I couldn't find any kind of manual for the game (some games have a pdf etc). Or does this all get explained ingame and I should just play a bit longer instead of wandering around leveling up?

Battles remaining is how many battles you have left until you stop meeting enemies in random encounters in that specific dungeon and/or area. Basically meaning you can explore without fear of coming into a random encounter after you've defeated the set number of battles. I might as well add that after the counter reaches zero, you can still battle against mobs if you choose "fight" from the menu.